Vacuum alignment and lattice artifacts: Wilson fermions

0.22 MB·

by  Maarten Golterman

Vacuum alignment and lattice artifacts: Wilson fermions
Maarten Goltermana and Yigal Shamirb
aDepartment of Physics and Astronomy
San Francisco State University, San Francisco, CA 94132, USA
5
1 bRaymond and Beverly Sackler School of Physics and Astronomy
0 Tel Aviv University, Ramat Aviv, 69978 ISRAEL
2

y
a
M ABSTRACT

3
Confinement in asymptotically free gauge theories is accompanied by the
1
spontaneous breaking of the global flavor symmetry. If a subgroup of the flavor

] symmetry group is coupled weakly to additional gauge fields, the vacuum state
t
a
tends to align such that the gauged subgroup is unbroken. Independently, a
l
- lattice discretization of the continuum theory typically reduces the manifest
p
e flavor symmetry, and, in addition, can give rise to new lattice-artifact phases
h
with spontaneously broken symmetries. Here, we study the interplay of these
[
two phenomena for Wilson fermions, using chiral lagrangian techniques. We
3
consider two examples: electromagnetic corrections to QCD, and a prototype
v
6 composite-Higgs model.
5
3
0
.
1
0
4
1
:
v
i
X
r
a
1
I. INTRODUCTION
Over the last few years, the computation of certain hadronic quantities using lattice
QCD has become so accurate that electromagnetic effects, while typically small, need to be
included in order to further improve on present errors [1]. A further reduction of the lattice
spacing is also needed in order to suppress competing discretization effects.
There may indeed be a real competition between electromagnetic effects and lattice ar-
tifacts: Both can have a non-trivial influence on the phase diagram of the lattice theory.
First, given a strongly interacting gauge theory, let us weakly couple a subgroup of the flavor
symmetry to dynamical gauge fields (“weak gauge fields,” for short). It was observed long
ago that the weak gauge fields can influence the symmetry-breaking pattern. Their coupling
to unbroken flavor generators tends to stabilize the vacuum, whereas the coupling to broken
generators tend to destabilize it, a phenomenon usually referred to as “vacuum alignment”
[2]. Furthermore, depending on the resulting alignment, some of the Nambu–Goldstone
bosons (NGBs) associated with the symmetry breaking may acquire a mass, thereby becom-
ing pseudo-NGBs. An example is the QED-induced mass splitting between the charged and
neutral pions in QCD.
Even without weak gauge fields, a non-trivial phase structure can also emerge at non-zero
lattice spacing. An example is the possible appearance of a so-called Aoki phase in two-
flavor QCD with Wilson fermions. Depending on details of the regularization, a phase can
appear in which isospin is spontaneously broken to a U(1) subgroup, alongside with parity
[3–5].
It is interesting to study what happens when both effects are at work. For instance, in
lattice QCD with two degenerate Wilson fermions, what would happen to the Aoki phase if
QED is turned on, or if all the isospin generators are coupled to weak gauge fields?
Similar questions arise beyond the realm of QCD. The existence of a light Higgs particle
has revived interest in composite Higgs models, in which a strongly coupled gauge theory
breaks its flavor symmetry dynamically at the TeV scale, producing a massless meson with
the quantum numbers of the Higgs among the NGBs associated with the breaking of the
symmetry. The flavor currents of this strongly interacting theory can be coupled to a
number of weak gauge fields, with the Standard Model’s electro-weak gauge fields among
them. Electro-weak symmetry breaking is then arranged to take place through the effective
potential generated for the NGBs of the strongly coupled theory by the weak dynamics. A
prototype example of such a theory is the “Littlest Higgs” model of Ref. [6]. In this theory
theflavorsymmetry groupisSU(5), spontaneously brokenbythestrongdynamics toSO(5).
Weak gauge fields are coupled to an [SU(2) U(1)]2 subgroup of SU(5), with the Standard
×
Model’s electro-weak gauge fields coupling to the diagonal subgroup of [SU(2) U(1)]2,
×
which is also a subgroup of the unbroken SO(5).
A basic tool used in the phenomenological literature is the non-linear sigma model de-
scribing the (pseudo-) NGBs (for recent reviews, see Refs. [7, 8]). Such a low-energy effective
theory requires an “ultraviolet completion.” In many cases, the underlying theory can be
taken to be a confining gauge theory, which, in turn, can be studied on the lattice. One can
then use numerical methods in order to determine the low-energy constants (LECs) relevant
for electro-weak physics. Since not only the precise values of the LECs, but even their signs
are usually outside the scope of the non-linear sigma model, their determination is crucial if
2
we are to confirm that the correct symmetry-breaking pattern indeed takes place.1 Again,
the question arises whether lattice artifacts might have an effect on the phase structure,
possibly distorting the alignment properties of the continuum theory.
In this article, we consider these questions in the context of strongly coupled lattice gauge
theories with Wilson fermions. The use of Wilson fermions means that axial symmetries are
explicitly broken by the discretization, and they are recovered only in the continuum limit.
In the two-flavor theory, this leads to the practical limitation that weak dynamical gauge
fields can be coupled to isospin generators only, and not to the axial generators.
In order to realize the SU(5)/SO(5) non-linear sigma model we envisage a confining
theory with 5 Weyl fermions in a real representation of the strong gauge group [2]. In
the continuum, this strongly interacting theory can equivalently be formulated in terms of
Majorana fermions. Transcribing the latter theory to the lattice is straightforward. But,
once again, if we use Wilson fermions, only the SO(5) flavor symmetry is preserved, because
it is vectorial in the Majorana formulation. The remaining symmetries (which generate the
coset SU(5)/SO(5)) are axial. They are explicitly broken by the Wilson mass term, again
to be recovered only in the continuum limit. On the lattice we thus consider only dynamical
weak gauge fields for subgroups of SO(5). As we will see, this is sufficient to gain access to
LECs of the low-energy effective theory that are of interest to phenomenology.
In Sec. II we will consider two-flavor QCD with Wilson fermions, and investigate what
happens if we gauge all isospin generators, or if we gauge only the U(1) subgroup for the
I component of the photon. We will consider the lowest-order pion effective potential,
3
containing terms linear in the quark mass, quadratic in the lattice spacing, and linear in
the fine-structure constant, assuming that these are all of a comparable magnitude. In
Sec. III we will then consider the SU(5)/SO(5) non-linear sigma model, with the weak
gauge fields those of the Standard Model group SU(2) U(1) , in a similar framework.
L Y
×
Because of the more complicated form of the effective potential, we will not be able to fully
explore the phase diagram that may arise from discretization effects. However, a quadratic
fluctuation analysis around the vacuum of the continuum theory will still lead to non-trivial
observations. The final section contains our conclusions, and a proof of vacuum alignment
for the continuum SU(5)/SO(5) theory is relegated to an appendix.
II. TWO-FLAVOR QCD WITH WILSON FERMIONS
Following Ref. [5], we start from the effective potential for the pions of two-flavor lattice
QCD with Wilson fermions,2
c c
V = 1 tr(Σ+Σ†)+ 2 tr(Σ+Σ†) 2 (2.1)
eff
− 4 16
= c σ +c σ2 , (cid:0) (cid:1)
1 2
−
in which
Σ = σ +i τ π , σ2 + π2 = 1 , (2.2)
a a a
a a
X X
1 For realistic studies of the phenomenologyof such models, the top-quark sector should also be taken into
account.
2 For reviews of chiral perturbation theory for QCD with Wilson fermions, see Refs. [9–11].
3
is the non-linear SU(2) matrix built out of the isospin triplet of pion fields π , with τ
a a
the three Pauli matrices. The parameter c is linear in the PCAC quark mass m, while
1
c is proportional to the square of the lattice spacing a.3 Higher order terms in the chiral
2
expansion of V will be neglected, since they do not qualitatively affect the phase diagram
eff
(unless at least one of the leading-order terms vanishes).
In the continuum limit, c = 0, and there is a first-order phase transition when c , i.e.,
2 1
the quark mass m, changes sign: the condensate Σ = Σ realigns from Σ = +1 for c > 0
0 0 1
h i
to Σ = 1 for c < 0. At non-zero lattice spacing, this conclusion does not change if
0 1
−
c < 0, because the c term in V is minimized for Σ = 1, irrespective of the sign of Σ .4
2 2 eff 0 0
±
Compared to the continuum theory, the difference is that for c < 0 the pion masses do not
2
vanish at the transition; instead, they are all degenerate, and of order √ c a.
2
− ∝
For c > 0, the minimum of V is reached at
2 eff
1 , c 2c ,
1 2
≥
σ = c1 , 2c < c < 2c , (2.3)
h i  2c2 − 2 1 2
1 , c 2c .
 1 2
− ≤ −
For c < 2c we find that σ < 1, which implies that π = 0. SU(2) isospin is
1 2 a
| | |h i| h i 6
spontaneously broken to a U(1) subgroup,5 and two of the three pions become massless as
the NGBs associated with this symmetry breaking. This region in the phase diagram is the
Aoki phase. Clearly, in order to probe the Aoki phase transition, the couplings c c have
1 2
∼
to be of the same magnitude. We may take the direction of symmetry breaking to point in
the τ direction, so that π± are the NGBs, while π0 is massive inside the Aoki phase. At
3
the phase boundaries c = 2c all three pions are degenerate and massless, even though
1 2
| |
c m does not vanish. In the continuum limit, c a2 0, and the Aoki phase shrinks
1 2
∝ ∝ →
to zero; the continuum limit at c = c = 0 yields QCD with two massless quarks.
2 1
Inside the Aoki phase of the lattice theory, parity is spontaneously broken as well. In the
continuum, if we take the vacuum Σ = 1, parity acts as Σ Σ†. Since the symmetry is
h i ± →
SU(2) SU(2) , any expectation value Σ SU(2) can be rotated to Σ = 1 using,
L R
× h i ∈ h i ±
e.g., an SU(2) transformation. Thus, if we would want to expand around an equivalent
L
vacuum Σ = 1, parity would merely take a more complicated form. By contrast, on the
h i 6 ±
lattice the axial symmetries are explicitly broken. Vacua with different values of σ are
h i
inequivalent, and, for any π = 0, parity is broken spontaneously.
a
h i 6
A. Gauging isospin
We now consider what happens if we gauge isospin, with a gauge coupling g weak enough
that we can analyze the effect on the phase diagram by considering the order-g2 correction
to V . We expect that non-trivial modifications of the scenarios reviewed above may occur
eff
when g2 c c , or, equivalently, g2 m/Λ (aΛ )2.
1 2 QCD QCD
∼ ∼ ∼ ∼
3 Terms linear in the lattice spacing break the symmetry in exactly the same way as the term linear in
the quark mass, and are thus absorbed into the term proportional to c . Since both c and c (and
1 1 2
c in Sec. IIB below) have mass dimension equal to four, appropriate powers of Λ will always be
3 QCD
understood.
4 In the large-N limit, c <0 is excluded [12], but at finite N both signs are possible.
c 2 c
5 For the reason that the Vafa–Witten theorem [13] does not apply inside the Aoki phase, see Ref. [5].
4
In order to find the order-g2 part of V we proceed as follows. The lowest order chiral
eff
effective action contains a term
f2
= tr (D Σ)†D Σ , (2.4)
µ µ
L 8
(cid:0) (cid:1)
where f is the pion decay constant in the chiral limit, and
D Σ = ∂ Σ+ig[V ,Σ] , (2.5)
µ µ µ
with V = V τ /2 the isospin gauge field.6 Upon working out the non-derivative part
µ a µ,a a
of ,
L P g2f2
tr V2 V ΣV Σ† , (2.6)
4 µ − µ µ
we see, first of all, that the weak gauge (cid:0)fields V remain(cid:1)massless on the isospin-symmetric
µ
vacua Σ = 1. Furthermore, integrating over the weak gauge fields, we find the leading
0
±
order contribution to the effective potential (2.1):7
g2c
∆V = 3 tr τ Στ Σ† , (2.7)
eff a a
− 8
a
X (cid:0) (cid:1)
in which c is independent of g2 to leading order. From Ref. [14] we know that c > 0. Using
3 3
Eq. (2.2), we find for the effective potential
V +∆V = c σ +(c g2c )σ2 +constant . (2.8)
eff eff 1 2 3
− −
The effect of the weak gauge fields V on the phase diagram is very simple: the parameter
µ
c gets shifted to c g2c . If c < 0, the transition when c goes through zero remains first
2 2 3 2 1
−
order. Even in the continuum limit, when c = 0, all pions acquire a mass proportional to
2
g2c g. If c > 0, the Aoki transition changes into a first-order transition when the
3 2
∝
lattice spacing becomes small enough such that c < g2c . In other words, the Aoki phase
2 3
p
gets pushed away from the continuum limit.
B. Coupling the photon
The situation changes when we restrict the gauge field to V = A Q, with Q =
µ µ
diag(2/3, 1/3) = 1/6 + τ /2, and g = e, the electric charge, so that A is the photon
3 µ
−
field. In that case, the shift in the effective potential becomes
e2c
∆Vem = 3 tr τ Στ Σ† , (2.9)
eff − 8 3 3
(cid:0) (cid:1)
with the same coefficient c as in Eq. (2.7). Using Eq. (2.2) again,
3
e2c
V +∆Vem = c σ +c σ2 3 (σ2 +π2) . (2.10)
eff eff − 1 2 − 2 3
6 The gauging of the vector symmetries leads to explicit breaking of the axial symmetries.
7 The effective potential due to the weak gauge fields always has a similar form, even if some of the
weakly gauged symmetries are spontaneously broken. The reason is that the gauge bosons’ mass will be
proportional to gf, and thus gauge-field mass effects only show up in the effective potential at order g4.
5
Again, the analysis of this effective potential is very simple. Since c > 0, minimizing the
3
effective potential requires that σ 2 + π 2 = 1, i.e., π = π = 0, irrespective of the
3 1 2
h i h i h i h i
values of c and c . If c < 0, σ = 1 depending on the sign of c , the phase transition is
1 2 2 1
h i ±
first order, and takes place at c = 0. The term proportional to c raises the charged pion
1 3
mass relative to the neutral pion mass [15].
If c > 0, and c < 2c so that we are in the Aoki phase, again σ = c /(2c ) as in
2 1 2 1 2
| |
Eq. (2.3), and therefore
c2
π = 1 1 . (2.11)
h 3i s − 4c2
2
Isospin is explicitly broken by the coupling to QED, but parity is spontaneously broken in
the Aoki phase, and there still is a second order phase transition. Inside the Aoki phase,
the pion masses are
m2 = e2c f−2 , (2.12)
± 3
c2
m2 = 2c 1 1 f−2 .
0 2 − 4c2
(cid:18) 2(cid:19)
Wesee that, depending ontherelative size ofthe parametersc , c ande2c , the neutral pion
1 2 3
might even be heavier than the charged pion, even though in the continuum limit Witten’s
inequality implies that this can never be the case [14]. The reason is that now we have a
competition: electromagnetic effects increase the charged pion mass relative to the neutral
pion mass; whereas the lattice artifacts that give rise to the breaking of isospin in the Aoki
phase create an opposite effect, since the charged pions are the NGBs of this symmetry
breaking.
III. LITTLEST HIGGS
In this section we present an analysis of the Littlest Higgs model of Ref. [6]8 that parallels
what we did for QCD in Sec. II. First, we very briefly review the necessary ingredients of
this theory in the continuum, in Sec. IIIA, including the coupling to the Standard Model
gauge fields. We next consider the Aoki phase for this theory, without the weak gauge
fields, in Sec. IIIB. In Sec. IIIC we then consider the competition between the effective
potential generated by the weak gauge fields and that generated by lattice artifacts in the
determination of the phase diagram.
A. Littlest Higgs – continuum
Weconsider astronglycoupledgaugetheorywith5Weyl fermionsinareal representation
of the (unspecified) strong gauge group. This theory has an SU(5) flavor symmetry which
we assume to be broken to SO(5) by a bilinear fermion condensate, resulting in 14 NGBs
parametrizing the coset SU(5)/SO(5). In order to construct the effective theory for these
NGBs, we introduce the non-linear field
Σ = exp(iΠ/f)Σ exp(iΠT/f) = exp(2iΠ/f)Σ , (3.1)
0 0
8 See also Ref. [8] for a review.
6
with Σ = Σ given by9
0
h i
0 0 1 0 0
0 0 0 1 0
 
Σ = 1 0 0 0 0 . (3.2)
0
0 1 0 0 0 
 
0 0 0 0 1 
 
 
Since the bilinear fermion condensate is symmetric in its SU(5) indices, so is Σ. Therefore,
Σ transforms into UΣUT with U SU(5), and this leads to the form (3.1) for Σ in terms of
∈
the “pion” field Π, which satisfies Σ ΠT = ΠΣ . The generators T of the unbroken SO(5)
0 0
obey the relation Σ TT = TΣ .
0 0
−
TheStandardModelSU(2) gaugefieldsW arecoupledtoanSU(2)subgroupofSO(5)
L µa
generated by [6]
1τ 0 0
2 a
Q = 0 1τT 0 , a = 1, 2, 3 , (3.3)
a  −2 a 
0 0 0
 
where again τ are the Pauli matrices. The SU(2) generated by the Q is an invariant
a a
subgroup of the SO(4) group defined by embedding its elements in the upper-left 4 4
×
block of the SO(5) matrices.
The leading-order effective potential for the Σ field, obtained by integrating over the W
fields, is given by
V = g2C tr (ΣQ Σ∗Q∗) , (3.4)
weak w a a
where a sum over a is implied. The low-energy constant C is analogous to the constant c
w 3
in Eq. (2.7), and it is positive, as we show in App. A, using the relevant result of Ref. [14].
In Ref. [6] more weak gauge fields are coupled to a subgroup of SU(5) in order to obtain the
“collective” symmetry breaking typical of little-Higgs models. However, the primary goal
of a lattice investigation of this theory would presumably be the determination of the LEC
C , which can be probed using any subgroup of SU(5), such as, for instance, the SU(2)
w
group we introduced in Eq. (3.3). As we explain below, this allows us to maintain all gauged
symmetries (strong and weak) on the lattice if we choose to work with Wilson fermions.
In Eq. (3.4), the minimum value for the trace, 3, is attained for Σ = Σ . Therefore the
0
−
vacuum is aligned, i.e., the W fields do not move the vacuum away from Eq. (3.2). The
potential V is invariant under the SO(4) subgroup defined above: If we transform Σ
weak
→
UΣUT with U SO(4), we see that this is equivalent to keeping Σ fixed, while transforming
∈
Q UTQ U∗ = R Q inside the trace, with R in the fundamental representation of
a a ab b
→
SO(3). Here we used that the Q generate an invariant subgroup of SO(4). Using that
a
R R = (RTR) = δ the invariance follows.
ab ac bc bc
We may also introduce the hypercharge weak gauge field, which gauges the U(1) symme-
try generated by [6]
1
Y = diag(1,1, 1, 1,0) . (3.5)
2 − −
This breaks the SO(4) symmetry explicitly to SU(2) U(1) , with SU(2) the group to
L Y L
×
which the W fields couple. The new contribution to the effective potential is
V = g′2C tr (ΣYΣ∗Y) , (3.6)
Y w
9 Relative to Ref. [6] we interchanged the 3rd and 5th rows and columns in the form for Σ .
0
7
where the constant C is the same as in Eq. (3.4), and g′ the hypercharge gauge coupling.
w
In order to move to the lattice, the strongly interacting theory is first reformulated in
terms of Majorana fermions instead of Weyl fermions. Now, because the fermions transform
in a real representation of the strong gauge group, a gauge-invariant fermion mass term can
be added to the theory, breaking SU(5) SO(5) softly. Going to the lattice using Wilson
→
fermions, it is then straightforward to augment this local mass term with a Wilson mass
term as well, in order to avoid species doublers. The exact flavor symmetry of the lattice
theory is therefore just SO(5), regardless of the fermion mass. We expect the full SU(5)
symmetry toberestored inthecontinuum limit, provided thatthesingle-site Majoranamass
is tuned appropriately. These features are, of course, completely analogous to the usual case
of Wilson-Dirac fermions.
On the lattice, before any weak gauge fields are coupled to the flavor currents and for a
large-enough positive quark mass, the fermion condensate will be proportional to the unit
matrix(seeSec. IIIBbelow). Anticipatingthis, itisconvenient toreformulatethe(massless)
continuum effective theory such that this is also the case there. Starting from Eq. (3.2) it is
straightforward to find an element U of SU(5) such that
Σ′ = UΣ UT = 1 . (3.7)
0 0
We also have to transform the generators Q and Y to the new basis, defining
a
W UQ U† , X UYU† . (3.8)
a a
≡ ≡
Since Σ′ isproportionalto theunit matrix, theW andX areanti-symmetric andhermitian,
0 a
and thus purely imaginary. The potential V +V can be written as
weak Y
V +V = g2C tr (ΣW Σ∗W ) g′2C tr (ΣXΣ∗X) . (3.9)
weak Y w a a w
− −
After adding V +V to the effective theory, the complete vacuum manifold is the U(1)
weak Y
circle generated by T = diag(1,1,1,1, 4).10
−
For Majorana (equivalently, Weyl) fermions there are no separate C and P symmetries,
only a CP symmetry. The role of CP parallels that of parity in the two-flavor theory of
Sec. II. If we expand the non-linear field around the unit matrix, CP acts on the pion
field as Π Π. Since the vacuum manifold contains the unit matrix, it follows that CP
→ −
symmetry is unbroken in the continuum theory.
B. Littlest Higgs – lattice artifacts
In this subsection, we turn off the weak gauge fields, and consider only the strongly
coupled theory on the lattice, using Wilson-Majorana fermions.
The construction of the effective potential representing the effects of a quark mass and
lattice artifacts to order a2 for the SU(5)/SO(5) effective theory is very similar to the
construction for the (SU(2) SU(2) )/SU(2) case reviewed in Sec. II. The only difference
L R
×
10 SincewegaugeonlythegeneratorsQ =Qa+Qa andY =Y +Y ofRef.[6],the Higgsfieldcomponents
a 1 2 1 2
of Π pick up a mass, see Sec. IIIC below.
8
is that more invariants proportional to a2 exist, so that now the effective potential becomes
[16]
c c c c
V = 1 tr(Σ+Σ†)+ 2 tr(Σ+Σ†) 2 3 tr(Σ Σ†) 2 + 4 tr Σ2 +Σ†2 , (3.10)
Aoki
− 2 4 − 4 − 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
in which c is proportional to the (subtracted) quark mass, and c are all proportional to
1 2,3,4
a2.11 There is no symmetry relating the theory with c > 0 to that with c < 0, because no
1 1
non-anomalous transformation exists that relates the two. We will therefore mostly limit
ourselves to the choice c 0 in this article.
1
≥
On our new basis the pion field Π in Eq. (3.1) is real and symmetric, and can thus be
diagonalized by an SO(5) transformation. It follows that in order to find the minimum of
V we may choose Σ in Eq. (3.10) to be diagonal,
Aoki
Σ = diag eiφ1,eiφ2,eiφ3,eiφ4,eiφ5 , (3.11)
(cid:0) (cid:1)
subject to the constraint
5
φ = 0 (mod 2π) . (3.12)
i
i=1
X
Substituting this into Eq. (3.10) yields
2 2
V = c cosφ 2c cos2φ +c cosφ +c sinφ . (3.13)
Aoki 1 i 4 i 2 i 3 i
− −
! !
i i i
X(cid:0) (cid:1) X X
This is not easily minimized, so we will begin with simplifying V by omitting the double-
Aoki
trace terms, i.e., by setting c = c = 0. Even with only c and c , the minimization of V
2 3 1 4 Aoki
will not be a simple task, because of the constraint (3.12).
For c < 0, the minimum is at φ = 0, as in the case of Sec. II, and the pseudo-NGBs
4 i
remainmassive inthelimit c 0, aslongasc = 0; their massisproportionalto√c 4c .
1 4 1 4
→ 6 −
For c > 0, we will proceed in several steps. First we prove that for c > 4c the solution
4 1 4
is again φ = 0, so that no symmetry is spontaneously broken. We will then analyze the
i
case that c = 4c 2ǫ with ǫ > 0 small, as well as the case that c = ǫ is small. Since we
1 4 1
−
may take c to set the overall scale of V , we will set c = 1 in most of the rest of this
4 Aoki 4
subsection.
The potential V is extremized if
Aoki
sinφ (c 4cosφ ) = λ , (3.14)
i 1 i
−
where λ is a Lagrange multiplier enforcing the constraint. First, let us ignore the constraint,
which is equivalent to setting λ = 0. Then, for c > 4, Eq. (3.14) implies that φ = 0, if
1 i
we also demand the solution to be the minimum of V . Since this solution satisfies the
Aoki
constraint (3.12), we have found the solution we are looking for. Also, since there is only
one minimum for c > 4, it follows by continuity that the same is true at c = 4. Therefore,
1 1
if a phase transitions occurs at c = 4, this phase transition is second order.
1
11 For SU(2), the last three terms collapse to the one term in Eq. (2.1).
9
Next, we consider c = 4 2ǫ, with ǫ > 0 small. Since only a continuous phase transition
1
−
may take place, φ will be small as well, and we thus expand the left-hand side of Eq. (3.14)
i
to order φ3:
i
φ ǫ+φ2 = λ/2 . (3.15)
i − i
From this, it follows that for any tripl(cid:0)e i,j,k, i(cid:1)f φi is equal to neither φj nor φk, then
φ2 +φ φ +φ2 = φ2 +φ φ +φ2 = ǫ . (3.16)
i i j j i i k k
It follows that either φ = φ , or φ = φ φ . This provides us with a finite list of options
k j k i j
− −
to check, and we find that V is minimized for
Aoki
Σ = Σ (4 2ǫ) = exp[idiag(φ,φ,φ, 3φ/2, 3φ/2)] , (3.17a)
0
− − −
9
φ2 = ǫ . (3.17b)
7
Indeed, a second order phase transition takes place at c = 4, with, below that value, a
1
symmetry-breaking patternSO(5) SO(3) SO(2). Inaddition, CPsymmetry, Σ Σ∗, is
→ × →
spontaneously broken as well. We note that the solution (3.17) cannot be rotated to Σ = 1,
0
because on the lattice the SU(5) transformation that would do this is not a symmetry.
We now turn to the case that c = 0. If φ is a solution of Eq. (3.14), i.e., sin2φ = λ/2,
1 0 0
−
then all possible solutions are
φ = φ , φ = π/2 φ , φ = π +φ , φ = 3π/2 φ . (3.18)
i 0 i 0 i 0 i 0
− −
Going through all possibilities for choosing the φ , i = 1,...,5 from this list, and demanding
i
thatanysuchchoicesatisfies theconstraint(3.12), yieldsthreedegenerateminimaforc = 0:
1
(1)
Σ = Σ (0) = exp[(2πi/5)diag(1,1,1,1,1)] , (3.19a)
0
(2)
Σ = Σ (0) = exp[(2πi/5)diag(1,1,1, 3/2, 3/2)] , (3.19b)
0 − −
(3)
Σ = Σ (0) = exp[(2πi/5)diag(1, 3/2, 3/2, 3/2, 3/2)] . (3.19c)
0 − − − −
Next, let us consider small c = ǫ. Once again, since the three global minima at c = 0
1 1
are discrete, this can at most lead to a small shift δφ away from 2π/5 or 3π/5 for each i.
i
−
Expanding V , we find
Aoki
5 1
V(1) = 3 √5 ǫ √5 1 + 1+√5 δφ2 , (3.20a)
Aoki 4 − − − 2 i
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)Xi
5 ǫ 1
V(2) = 3 √5 √5 1 + 1+√5 δφ2 , (3.20b)
Aoki 4 − − 5 − 2 i
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)Xi
5 3ǫ 1
V(3) = 3 √5+ √5 1 + 1+√5 δφ2 , (3.20c)
Aoki 4 − 5 − 2 i
(cid:18) (cid:16) (cid:17)(cid:19) (cid:16) (cid:17)Xi
wherethesuperscriptonV referstowhichsolutioninEq.(3.19)weareexpandingaround.
Aoki
We have expanded to quadratic order in δφ , dropping terms of order ǫδφ2, and we have
i i
(1)
used that δφ = 0 because of Eq. (3.12). For small c = ǫ > 0 the first minimum, Σ (0),
i i 1 0
is the absolute minimum, and, since the coefficients of the δφ2 terms in Eq. (3.20) are always
i
P
10